Modified Cyclotron Frequency Research Articles

Coupled-oscillator model to analyze the interaction between a quartz resonator and trapped ions

The novel application of a piezoelectric quartz resonator for the detection of trapped ions has developed in the observation of the quartz-ions interaction under non-equilibrium conditions, opening new perspectives for high-sensitive motional frequency measurements of radioactive particles. Energized quartz crystals have (long) constant-decay times in the order of milliseconds, permitting the coherent detection of charged particles within short times. In this publication we develop in detail a model governing the interaction between trapped $^{40}$Ca$^+$ ions and a quartz resonator connected to a low-noise amplifier. We apply this model to experimental data and extract relevant information like the coupling constant $g=2\pi \times 1.449(2)$~Hz and the ions' modified-cyclotron frequency in our 7-tesla Penning trap. The study on the latter is specially important for the use of this resonator in precision Penning-trap mass spectrometry. The improvement in sensitivity can be accomplished by increasing the coupling constant through the quality factor of the resonator. This can develop in the use of the hybrid quartz-ion system for other applications.

On magnetic monopole and Dirac quantization condition in the minimal length formalism

In this article, we study the effects of minimal length corrections on the cyclotron acceleration and radiation of a charged particle moving in a magnetic monopole field. Using the modified Hamiltonian describing the dynamics of a charged particle moving in a magnetic monopole field, we derive the modified cyclotron frequency and modified acceleration. Also, we derive the modified power for the cyclotron radiation using the modified acceleration. We find that all modified quantities are dependent on the deformation parameter α. In addition, we find that the modified electromagnetic angular momentum and magnetic charges are not quantized as an integer multiple; thus, we establish that the modified Dirac quantization condition and the usual Dirac quantization condition is just the term of zero order in α. The modified Dirac quantization condition allows fractional charges that arise from vacuum fluctuations. In the Born approximation, we derive the modified scattering cross-section and then we estimate the upper bound on the minimal length.

Sideband cooling of the radial modes of motion of a single ion in a Penning trap

Doppler and sideband cooling are long standing techniques that have been used together to prepare trapped atomic ions in their ground state of motion. In this paper we study how these techniques can be extended to cool both radial modes of motion of a single ion in a Penning trap. We numerically explore the prerequisite experimental parameters for efficient Doppler cooling in the presence of an additional oscillating electric field to resonantly couple the radial modes. The simulations are supported by experimental data for a single $^{40}$Ca$^+$ ion Doppler cooled to $\sim$100 phonons in both modes at a magnetron frequency of 52 kHz and a modified cyclotron frequency of 677 kHz. For these frequencies, we then show that mean phonon numbers of $0.35(5)$ for the modified cyclotron and $1.7(2)$ for the magnetron motions are achieved after 68 ms of sideband cooling.

Taming Hamiltonian systems with balanced loss and gain via Lorentz interaction: general results and a case study with Landau Hamiltonian

The kinetic energy term of Hamiltonian systems with balanced loss and gain is not semi-positive-definite, leading to instabilities at the classical as well quantum level. It is shown that an additional Lorentz interaction in the Hamiltonian allows the kinetic energy term to be semi-positive-definite and thereby, improving the stability properties of the system. Further, a consistent quantum theory admitting bound states may be obtained on the real line instead of Stoke wedges on the complex plane. The Landau Hamiltonian in presence of balanced loss and gain is considered for elucidating the general result. The kinetic energy term is semi-positive-definite provided the magnitude of the applied external magnetic field is greater than the magnitude of the ‘analogous magnetic field’ due to the loss-gain terms. It is shown that the classical particle moves on an elliptical orbit with a cyclotron frequency that is less than its value in absence of the loss-gain terms. The quantum system share the properties of the standard Landau Hamiltonian, but, with the modified cyclotron frequency. It is shown that the Hall current has non-vanishing components along the direction of the external uniform electric field and to its transverse direction. The Pauli equation in presence of balanced loss and gain is shown to be supersymmetric.

Thomson backscattering in combined uniform magnetic and envelope modulating circularly polarized laser fields

The Thomson backscattering spectra in combined uniform magnetic and cosine-envelope circularly-polarized laser fields are studied in detail. With an introduction of the envelope modulation, the radiation spectra exhibit high complexity attributed to the strong nonlinear interactions. On the other hand, four fundamental laws related to the scale invariance of the radiation spectra are analytically revealed and numerically validated. They are the laws for the radiation energy as the $6$th power of the motion constant exactly, also as the approximate negative $6$th power with respect to the initial axial momentum and laser intensity in a certain of conditions, respectively, and finally an important self-similar law, i.e., when the circular laser frequency, the envelope modulation frequency, and the modified cyclotron frequency are simultaneously increased by a factor, the radiation energy will be increased by the second power of that factor without changing the shape of the spectrum. With the application of these laws, especially the last one, a much higher radiation energy can be obtained and the harmonic at which the maximum radiation occurs can be precisely tuned without changing its amplitude. These findings provide a possible way to advance radiation technology in many fields such as medicine, communications, astrophysics, and security.

Penning trap with an inclined magnetic field.

A modified Penning trap with a spatially uniform magnetic field B inclined with respect to the axis of rotational symmetry of the electrodes is considered. The inclination angle can be arbitrary. Canonical transformation of phase variables transforming the Hamiltonian of the considered system into a sum of three uncoupled harmonic oscillators is found. We determine the region of stability in space of two parameters controlling the dynamics: the trapping parameter κ and the squared sine of the inclination angle ϑ0. If the angle ϑ0 is smaller than 54°, a charge occupies a finite spatial volume within the processing chamber. A rigid hierarchy of trapping frequencies is broken if B is inclined at the critical angle: the magnetron frequency reaches the modified cyclotron frequency while the axial frequency exceeds them. Apart from this resonance, we reveal the family of resonant curves in the region of stability. In the relativistic regime, the system is not linear. We show that it is not integrable in the Liouville sense. The averaging over the fast variable allows to reduce the system to two degrees of freedom. An analysis of the Poincaré cross-sections of the averaged systems shows the regions of effective stability of the trap.

Landau level transitions in doped graphene in a time dependent magnetic field

The aim of this work is to describe the Landau level transitions of Bloch electrons in doped graphene with an arbitrary time dependent magnetic field in the long wavelength approximation. In particular, transitions from the m Landau level to the m±1 and m±2 Landau levels are studied using the time dependent perturbation theory. Time intervals are computed in which transition probabilities tend to zero at a low order in the coupling constant. In particular, Landau level transitions are studied in the case of Bloch electrons traveling in the direction of the applied magnetic force and the results are compared with classical and revival periods of electrical current in graphene. Finally, current probabilities are computed for the n=0 and n=1 Landau levels showing expected oscillating behavior with modified cyclotron frequency.

One- and two-pulse quadrupolar excitation schemes of the ion motion in a Penning trap investigated with FT-ICR detection

In a Penning ion trap the interconversion between the radial motional modes of stored particles can be accomplished by applying one- and two-pulse (Ramsey) azimuthal quadrupolar radio frequency fields. In this work the interaction of ions with the excitation fields has been probed by Fourier transform ion cyclotron resonance (FT-ICR) detection. A theoretical description of this interaction is derived by use of a quasi-classical coherent state and the interconversion of modes is interpreted in a quantum-mechanical context. The dipolar-detection FT-ICR signal at the modified cyclotron frequency has been studied as a function of the interaction parameters such as excitation frequency, amplitude and duration and is compared with the theoretical results.

Phase-Sensitive Cyclotron Frequency Measurements at Ultralow Energies

A novel technique for a direct and coherent measurement of the modified cyclotron frequency of an ion in a Penning trap at energies close to the thermal cooling limit is presented. This allows a rapid and both precise and accurate determination of the free-space cyclotron frequency in real Penning traps despite the existence of electric and magnetic field imperfections and relativistic shifts. The demonstrated performance paves the way for considerably improved bound-state g-factor measurements on the 10ppt level and mass measurements in the 1ppt range and possibly below.

Nonlocal theory of the spectra of modified ion cyclotron oscillations in a charged plasma produced by gas ionization

The problem of the stability of a charged plasma cylinder in crossed longitudinal magnetic and radial electric fields is considered under the assumption that the plasma electrons are magnetized and distributed uniformly in space and that the plasma ions have a low density and move without collisions. By using the Vlasov and Poisson equations for the electric and magnetic fields of arbitrary strengths and by expanding the radial eigenfunctions in Bessel functions, a dispersion relation is obtained for the modified ion cyclotron frequencies. The dispersion relation obtained is solved for the case of hot plasma electrons. At low ion densities, the oscillation spectra are stable and are described by the families of dispersion curves lying closely around the harmonics of the modified cyclotron frequency (including the zeroth harmonic). At higher ion densities, the dispersion curves for the frequencies of the lowest radial oscillation modes in neighboring families can intersect, leading to instability. The maximum instability growth rate can be several tenths of the modified ion cyclotron frequency.

Ion dynamics in perturbed quadrupole ion traps

We use the results of a Lagrangian formulation of the dynamics of ions in a Penning trap to calculate the motional frequencies of the ions as a function of the trapping parameters. We then add realistic perturbations to the ideal trapping fields in the formalism and deduce the effects of these perturbations on the ion motion. For an ideal trap there exist values of the applied trapping fields that result in a degeneracy in the ion oscillation frequencies associated with different types of motion. We show that this motional frequency degeneracy is lifted by the inclusion of the effect of a tilt of the trap axis with respect to the axis of the applied magnetic field, leading to an ``avoided crossing'' between the oscillation frequencies. We calculate typical ion orbits for trap parameters that give oscillation frequencies near the avoided crossing between the axial and modified cyclotron frequencies. We generalize the analysis to include the motion of ions in a combined (Penning-Paul) trap and perform an experiment to test the predictions of the theory for a degeneracy between the modified cyclotron frequency and the axial frequency for ${\mathrm{Mg}}^{+}$ ions held in a tilted combined trap. The oscillation frequencies are measured for a range of tilt angles using a photon-photon correlation technique. There is good agreement between the experimental results and the theoretical predictions. The method we describe may prove to be a useful means by which trap imperfections can be identified and subsequently removed.

Width effect on the magnetoconductivity of 2D electron systems

Based on a parabolic well model, it is shown that the magnetoconductivity of narrow 2D electron systems is characterized by a modified cyclotron frequency and a crossover parameter which is a ratio of the cyclotron frequency to the curvature of the parabolic potential. Explicit results are derived for systems with finite width under the assumption that quantum oscillations are small.